As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples here I added them as answers so they may be voted up and down with the rest.

This is a community-wiki question: One colloquial statement and its mathematical meaning per answer please!

52 Answers
52

A drunk man will find his way home, but a drunk bird may get lost forever.

This encodes the fact that a 2-dimensional random walk is recurrent (appropriately defined for either the discrete or continuous case) whereas in higher dimensions random walks are not. More details can be found for instance in this enjoyable blog post by Michael Lugo.

This was Kac's famous way of asking whether the shape of a two-dimensional domain could be reconstructed from the spectrum of the Laplacian on that domain. (The answer, by the way, is "no", at least if one allows the domain to have corners.)

This should also follow from Brouwer's fixed-point theorem, right? If so, it's the best statement of Brouwer's fixed-point theorem I've ever heard, and I'll definitely be using it in the future!
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VectornautNov 17 '09 at 19:02

3

You need Banach's fixed point for the uniqueness, but the existence follows Brouwer's fixed point.
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Josiah SugarmanDec 10 '09 at 15:59

6

i actually did this in class to demonstrate the fixed point theorem. there was a satisfying gasp when I picked up my lecture notes and scrunched them :)
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Suresh VenkatDec 10 '09 at 20:36

9

With this formulation one may say that even the proof somewhat made its way thorugh and reached literature. In Borges "Partial Enchantments of the Quixote" a story contained in Other Inquisitions, an apocryphal quote states that in a perfect map, a copy of the map should be contained, and such copy would contain another copy of the map and so on at infinity, which is basically the proof of the Theorem, if you add to it existence of the limit...
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Nicola CiccoliJul 5 '12 at 7:24

By the way, Motzkin's remark, which is in a paper in J. Comb. Theory 3:244-252 (1967) made an interesting comparison between disorder in physics and combinatorics: "Whereas the entropy theorems of probability and mathematical physics imply that, in a large universe, disorder is probable, certain combinatorial theorems imply that complete disorder is impossible."
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John StillwellJan 31 '10 at 6:04

"X is the spitten image of Y" is an informal phrase meaning "X looks just like Y". "X is the spitting image of Y" is an entirely equivalent phrase; in some dialects, the pronunciation is even the same. As far as I know, neither is the "right spelling"; the phrase rarely appears in written English.
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jasomillMay 10 '11 at 10:48

This summarizes a paper (of the same name) by Li and Yorke. The full statement of the main theorem is that if a continuous transformation of an interval has a point whose orbit has length three, then there exist points whose orbits are completely chaotic (in addition to points with orbits of every other possible finite length).

I learned it in an Israeli high school from a Russian teacher and he called it two policemen and a drunk. So the drunk is between the two policmen who are going to the station.
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Yiftach BarneaApr 19 '11 at 7:19

10

And then there is the running joke of calling it the "three policemen" theorem because the drunk is a policeman as well.
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darij grinbergApr 19 '11 at 8:29

This refers to the fact that "sound" propagation in arbitrary dimensions (using the wave equation on an initial pulse) is only possible in odd-dimensional spaces (in even dimensions there's an ever-lasting echo), and that dispersion-free propagation only happens for dimension three.

See this paper where I saw the quote: part one, part two, although the fact itself seems to be known for ages (I have no access but an old paper by Balazs seems relevant too), possibly this even goes back to Petrovskii according to an interview of Arnold (see p434).

I think this is a dangerous metaphor: it encourages the misconception that "a space is compact iff it has a finite open cover".
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Rasmus BentmannMay 10 '11 at 8:04

2

This is attributed to Weyl in Wilson Sutherland's text Introduction to Metric and Topological Spaces (section 5.2). Sutherland gives it as "If a city is compact, it can be guarded by a finite number of arbitrarily near-sighted policemen". Exercise 5.10.15 asks you to make precise, and discuss the accuracy of, Weyl's statement.
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Tom LeinsterAug 5 '12 at 21:40

Two plus two is four. You can prove that two plus two is four. You can prove that you can prove that two plus two is four. And you can prove that you can prove that you can prove that two plus two is four, and so on.

Two plus two is not five. You can prove that two plus two is not five. You can't prove that two plus two is five, or else math is a lot of bunk. But, if math is not a lot of bunk, you can't prove that you can't prove that two plus two is five.

This no-hair theorem "postulates that all black hole solutions of the Einstein-Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum".

According to the Wikipedia page on this catchphrase (yes, Wikipedia has a "trust, but verify" page), it originates in a Russian proverb, as Reagan himself introduced it.
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Noam D. ElkiesAug 1 '11 at 4:34

A variant (with different mathematical overtones): "Think globally, act locally". I had a button with this on it as an undergraduate, from a very non-mathematical source.
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Ravi VakilJan 31 '10 at 5:38

The ham sandwich theorem comes to mind: given n measurable sets in Rn, there is a hyperplane (i. e. an affine subspace of codimension 1) that bisects them all. I don't know of a colloquial way to state this, though.

You can just say: "a ham sandwich (two pieces of bread and one of ham) can be split in half with a single cut."
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RicardoNov 1 '09 at 14:16

1

Ricardo, that's true. But I feel like what's really remarkable is that this holds in all dimensions, and ham sandwich definitely brings to mind a three-dimensional picture.
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Michael LugoNov 1 '09 at 16:39

9

So just take an n-dimensional ham sandwich... :)
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Cam McLemanJul 5 '10 at 5:12